Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}6x-3y &= 1 \\ 7x-6y &= 2\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-2$ and the bottom equation by $1$ $\begin{align*}-12x+6y &= -2\\ 7x-6y &= 2\end{align*}$ Add the top and bottom equations. $-5x = 0$ Divide both sides by $-5$ and reduce as necessary. $x = 0$ Substitute $0$ for $x$ in the top equation. $6( 0)-3y = 1$ $-3y = 1$ $-3y = 1$ $y = -\dfrac{1}{3}$ The solution is $\enspace x = 0, \enspace y = -\dfrac{1}{3}$.